The Stacks project

Proposition 32.9.6. Let $f : X \to S$ be a morphism of schemes. Assume

  1. $f$ is of finite type and separated, and

  2. $S$ is quasi-compact and quasi-separated.

Then there exists a separated morphism of finite presentation $f' : X' \to S$ and a closed immersion $X \to X'$ of schemes over $S$.

Proof. Apply Lemma 32.9.5 and note that $X_ i \to S$ is separated for large $i$ by Lemma 32.4.17 as we have assumed that $X \to S$ is separated. $\square$

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